Element Stiffness Coefficients Research Articles

Three-dimensional modelling of shear keys in concrete gravity dams using an advanced grillage method

Contraction joint shear keys are resilient features of gravity dams that can be considered to increase the sliding safety factors or minimise seismic residual sliding displacements, allowing costly remedial actions to be avoided. This paper presents a novel, robust, and computationally efficient three-dimensional (3D) modelling and simulation strategy of gravity dams, using a series of adjacent cantilever beam elements to represent individual monoliths. These monoliths are interconnected in the longitudinal direction by 3D no-tension link elements representing the lumped shear key stiffness contributions at a particular elevation. The objective is to assess the shear key internal force demands, including the axial force, shear, and moment demands. Shear key demand-capacity ratios can then be assessed with related multi-axial failure envelopes. The 3D link element stiffness coefficients were derived from a series of 3D finite element (FE) solid models with a detailed representation of geometrical features of multiple shear keys. The results from the proposed method based on advanced grillage analysis show strong agreement with reference solutions from 3D FE solid models, demonstrating high accuracy and performance of the proposed method. The application of the proposed advanced grillage method to a dam model with two monoliths clearly shows the advantage of the proposed method, in comparison to the classical approach used in practise.

An Efficient and Rapid Numerical Quadrature to generate element matrices for quadrilateral and hexahedral elements in Functionally Graded Materials (FGMs)

This research paper introduces a robust, efficient and stabilized midpoint quadrature to generate element stiffness matrices for quadrilateral and hexahedral elements in finite element method (FEM). This new quadrature is established based on Passive Richardson extrapolation which has the inherent ability to stabilize and improve the accuracy of the extrapolated results. The new quadrature includes a simple artificial stabilizing function with Gauss one-point quadrature (1GP) as an anti-hourglass effect. For the first approximation, the 1GP (sampling point to be considered at origin (0,0)) is done to evaluate element matrices (thermal conductivity matrices, capacitance matrices etc.,) in the 4-square element ( [-2, 2] x [-2, 2] ) in the (ξ, η) plane. As a stabilizing function, sampling points are considered to be either at the center of the four element edges or center of the four quadrants in the reference square element with appropriate constant weighting functions. For hexahedral elements also, the element stiffness coefficients are calculated only in the center of the element and center of all the octants of the reference cubic element ([-2, 2] x [-2, 2] x [-2, 2]), the proposed method deals mostly with integer variables and sampling at (0,0,0), whereas Gauss quadrature uses lengthy float variables corresponding to irrational numbers in terms of decimals in the stiffness matrix calculations, thus consuming time. A number of standard example problems are demonstrated to assess the efficiency of the proposed method. The results of all the numerical examples are found to be in good agreement with the exact results and conventional Gauss quadrature results.

First-order zig-zag sublaminate plate theory and finite element model for laminated composite and sandwich panels

A refined laminated plate theory and three-dimensional finite element based on first-order zig-zag sublaminate approximations has been developed. The in-plane displacement fields in each sublaminate are assumed to be piecewise linear functions and vary in a zig-zag fashion through-the-thickness of the sublaminate. The zig-zag functions are evaluated by enforcing the continuity of transverse shear stresses at layer interfaces. This in-plane displacement field assumption accounts for discrete layer effects without increasing the number of degrees of freedom as the number of layers is increased. The transverse displacement field is assumed to vary linearly through-the-thickness. The transverse normal strain predictions are improved by assuming a constant variation of transverse normal stress in each sublaminate. In the computational model, each finite element represents one sublaminate. The finite element is developed with the topology of an eight-noded brick, allowing the thickness of the plate to be discretized into several elements, or sublaminates, where each sublaminate can contain more than one physical layer. Each node has five engineering degrees of freedom, three translations and two rotations. Thus, this element can be conveniently implemented into general purpose finite element codes. The element stiffness coefficients are integrated exactly, yet the element exhibits no shear locking due to the use of an interdependent interpolation scheme and consistent shear strain fields. Numerical performance of the current element is investigated for a composite armored vehicle panel and a sandwich panel. These tests demonstrate that the element is very accurate and robust.

An improved theory and finite-element model for laminated composite and sandwich beams using first-order zig-zag sublaminate approximations

A new beam finite element based on a new discrete-layer laminated beam theory with sublaminate first-order zig-zag kinematic assumptions is presented and assessed for thick and thin laminated beams. The model allows a laminate to be represented as an assemblage of sublaminates in order to increase the model refinement through the thickness, when needed. Within each sublaminate, discrete-layer effects are accounted for via a modified form of DiSciuva's linear zig-zag laminate kinematics, in which continuity of interfacial transverse shear stresses is satisfied identically. In the computational model, each finite element represents one sublaminate. The finite element is developed with the topology of a fournoded rectangle, allowing the thickness of the beam to be discretized into several elements, or sublaminates, if necessary, to improve accuracy. Each node has three engineering degrees of freedom, two translations and one rotation. Thus, this element can be conveniently implemented into general purpose finite-element codes. The element stiffness coefficients are integrated exactly, yet the element exhibits no shear locking due to the use of a consistent interdependent interpolation scheme. Numerical performance of the current element is investigated for an arbitrarily layered beam, a symmetrically layered beam and a sandwich beam with low and high aspect ratios. The comparisons of numerical results with elasticity solutions show that the element is very accurate and robust.

Development of simple, robust finite elements based on refined theories for thick laminated beams

An approach for developing accurate, simple and robust two-noded C 0 finite elements based on shear deformable and layerwise (zig-zag) laminated beam theories is presented. First, a new generalized form of high-order shear deformation and zig-zag laminate theories is described that allows the continuity requirement on the transverse deflection degree freedom to be reduced. An associated finite element model is then developed that employs the penalty function concept in conjunction with an interdependent element interpolation scheme, making the element very accurate and robust for application to thick and thin laminated beams. The element stiffness coefficients are exactly integrated without giving rise to shear locking, and a consistent force vector is derived. Comparison of numerical results using the refined laminate models with exact elasticity solutions demonstrates that the current model is capable of accurately simulating the response of thin and moderately thick laminates, with or without damage.

Explicit determination of element stiffness matrices in the hybrid stress method

AbstractThe hybrid stress method has demonstrated many improvements over conventional displacement‐based formulations. A main detraction from the method, however, has been the higher computatational cost in forming element stiffness coefficients due to matrix inversions and manipulations as required by the technique. By utilizing permissible field transformations of initially assumed stresses, a spanning set of orthonormalized stress modes can be generated which simplify the matrix equations and allow explicit expressions for element stiffness coefficients to be derived. The developed methodology is demonstrated using several selected 2‐D quadrilateral and 3‐D hexahedral elements.

Static and dynamic response of moderately thick laminated beams with damage

The bending and vibration response of thin and moderately thick laminated beams with ply damage or delaminations is studied using a new discrete-layer laminated beam finite element. The two-noded C° element is based on a new generalized form of the first-order zig-zag laminate theory, and has only four degrees of freedom per node, regardless of the number of layers in the laminate. The element formulation employs the penalty-function concept in conjunction with an interdependent element interpolation scheme, making the element very accurate and robust for application to thick and thin laminated beams. The element stiffness coefficients are exactly integrated without giving rise to shear locking, and a consistent force and mass matrix is derived. The compliant layer concept is employed to simulate delaminations. Comparison of numerical results using the current model with predictions of exact elasticity solutions demonstrates that the current model is capable of accurately simulating the response of thin and moderately thick laminates that contain damage. Additional comparisons with equivalent single-layer theories indicate that the layerwise construction and kinematics must be explicitly taken into account when modelling laminates whose layers have drastically different stiffness properties, as is the case when damage is present.

Analysis of the Localized Buckling in Composite Plate Structures with Application to Determining the Strength of Corrugated Fiberboard

This study examines the theory for elastic buckling of composite structures composed of long rectangular flat plate elements. Elements can have nonlinear, orthotropic stress-strain properties characteristic of paper. Element stiffness coefficients add together to form a global structure stiffness matrix. Two methods of element counting and node labeling represent both determinant plate structures and periodic structures. Structure loading is in the axis direction of the plates. The solution gives the local buckling strain in the weakest structure element, which may differ from the weakest independent plate element. A nonlinear finite element method is developed with the finite elements being the plate elements of the structure. The small number of finite elements needed to characterize periodic structures makes the analysis efficient for searching out optimum designs. Corrugated fiberboard is one of the arbitrary structures appropriate for the analysis.

A finite element model for shells based on the discrete Kirchhoff hypothesis

AbstractA ring type finite element model based on the discrete Kirchhoff hypothesis is developed and its feasibility for use in shell analysis is illustrated by determining the linear axisymmetric deformation behaviour of a conical shell loaded by lateral pressure. The displacement field is represented by a linear function of the meridional co‐ordinate. The effects of coupling between bending and extension are included and in this respect the analysis is an extension of earlier work by other investigators using this same model, where bending deformations only were considered. The results of the investigation indicate that the proposed model is capable of accurately described coupled bending and extension effects with a relatively small number of elements. In addition to the numerical results obtained several distinct advantages and disadvantages of this element are brought out. In particular, the element stiffness coefficients can be easily derived because of the simplicity of the assumed vector field. However, it appears that the price paid for such a simple vector field approximation is an undue amount of machine storage, even for a relatively small number of elements. Several suggestions are made for minimizing some of these difficulties.